By contradiction, such polynomial must satisfy \(P(A)P(A)^{\top } = I_p\) . Thus, \(PP^{\top }\)  is of degree 0, hence \(P\)  is of degree 0, and \(P\)  is constant. Therefore, we cannot have \(P(A) =X + AX + o(\Vert AX\Vert )\) .

Of course, in practice, most retractions are implemented using polynomial approximations (see e.g. [1]}, [2]}). The previous proposition simply shows that polynomials can only be approximations, and as a consequence, any retraction must involve some linear algebra more complicated than matrix multiplication.
